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Volume 12, Number 4

Volume 12, Number 4, 2007

Toporensky A. V.,  Tretyakov P. V.
Certain Aspects of Regularity in Scalar Field Cosmological Dynamics
Abstract
We consider dynamics of the FRW Universe with a scalar field. Using Maupertuis principle we find a curvature of geodesics flow and show that zones of positive curvature exist for all considered types of scalar field potential. Usually, phase space of systems with the positive curvature contains islands of regular motion. We find these islands numerically for shallow scalar field potentials. It is shown also that beyond the physical domain the islands of regularity exist for quadratic potentials as well.
Keywords: islands of regular motion, scalar field cosmology
Citation: Toporensky A. V.,  Tretyakov P. V., Certain Aspects of Regularity in Scalar Field Cosmological Dynamics, Regular and Chaotic Dynamics, 2007, vol. 12, no. 4, pp. 357-364
DOI:10.1134/S1560354707040016
García-Naranjo L.
Reduction of Almost Poisson Brackets for Nonholonomic Systems on Lie Groups
Abstract
We present a systematic geometric construction of reduced almost Poisson brackets for nonholonomic systems on Lie groups with invariant kinetic energy metric and constraints. Our construction is of geometric interest in itself and is useful in the Hamiltonization of some classical examples of nonholonomic mechanical systems.
Keywords: nonholonomic systems, almost Poisson brackets, hamiltonization, geometric reduction
Citation: García-Naranjo L., Reduction of Almost Poisson Brackets for Nonholonomic Systems on Lie Groups, Regular and Chaotic Dynamics, 2007, vol. 12, no. 4, pp. 365-388
DOI:10.1134/S1560354707040028
Gurfil P.,  Elipe A.,  Tangren W.,  Efroimsky M.
The Serret–Andoyer Formalism in Rigid-Body Dynamics: I. Symmetries and Perturbations
Abstract
This paper reviews the Serret–Andoyer (SA) canonical formalism in rigid-body dynamics, and presents some new results. As is well known, the problem of unsupported and unperturbed rigid rotator can be reduced. The availability of this reduction is offered by the underlying symmetry, that stems from conservation of the angular momentum and rotational kinetic energy. When a perturbation is turned on, these quantities are no longer preserved. Nonetheless, the language of reduced description remains extremely instrumental even in the perturbed case. We describe the canonical reduction performed by the Serret–Andoyer (SA) method, and discuss its applications to attitude dynamics and to the theory of planetary rotation. Specifically, we consider the case of angular-velocity-dependent torques, and discuss the variation-of-parameters-inherent antinomy between canonicity and osculation. Finally, we address the transformation of the Andoyer variables into action-angle ones, using the method of Sadov.
Keywords: nonlinear stabilization, Hamiltonian control systems, Lyapunov control
Citation: Gurfil P.,  Elipe A.,  Tangren W.,  Efroimsky M., The Serret–Andoyer Formalism in Rigid-Body Dynamics: I. Symmetries and Perturbations, Regular and Chaotic Dynamics, 2007, vol. 12, no. 4, pp. 389-425
DOI:10.1134/S156035470704003X
Bloch A. M.,  Gurfil P.,  Lum K. Y.
The Serret–Andoyer Formalism in Rigid-Body Dynamics: II. Geometry, Stabilization, and Control
Abstract
This paper continues the review of the Serret–Andoyer (SA) canonical formalism in rigid-body dynamics, commenced by [1], and presents some new results. We discuss the applications of the SA formalism to control theory. Considerable attention is devoted to the geometry of the Andoyer variables and to the modeling of control torques. We develop a new approach to Stabilization of rigid-body dynamics, an approach wherein the state-space model is formulated through sets of canonical elements that partially or completely reduce the unperturbed Euler–Poinsot problem. The controllability of the system model is examined using the notion of accessibility, and is shown to be accessible. Based on the accessibility proof, a Hamiltonian controller is derived by using the Hamiltonian as a natural Lyapunov function for the closed-loop dynamics. It is shown that the Hamiltonian controller is both passive and inverse optimal with respect to a meaningful performance-index. Finally, we point out the possibility to apply methods of structure-preserving control using the canonical Andoyer variables, and we illustrate this approach on rigid bodies containing internal rotors.
Keywords: nonlinear stabilization, Hamiltonian control systems, Lyapunov control
Citation: Bloch A. M.,  Gurfil P.,  Lum K. Y., The Serret–Andoyer Formalism in Rigid-Body Dynamics: II. Geometry, Stabilization, and Control, Regular and Chaotic Dynamics, 2007, vol. 12, no. 4, pp. 426-447
DOI:10.1134/S1560354707040041

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